How to copy keyframes from one armature to another
When an animation is created, Blender creates an object data called "action", that can be appended or linked to another object (in your case the object will be the rig).
So you have to import in the new file the action from the old file.
you can reach and rename your actions through the dopesheet, changing in the header of the window from "dopesheet" to "actions".
Of course - to work properly -  both rigs should have the same bones, same names, etc..
Check manuals about "actions" for more infos.